Anti-eikonal equation of an eigenmirror.

We call a surface that appears undistorted when viewed in a curved mirror an eigensurface and the mirror an eigenmirror. Such pairs are described by a first-order nonlinear partial differential equation of the form a0+a1ux+a2uy+a3uxuy+a4ux2+a5uy2=0, where ai=ai(x,y,u), which we call the anti-eikonal equation. We give examples of symbolic and numerical solutions, including pairs that are geometrically congruent. Ray tracing simulations are included that visually confirm the unusual properties of these surfaces.